Hadamard gap series in weighted-type spaces on the unit ball

TL;DR

This paper characterizes when Hadamard gap series belong to weighted-type spaces on the unit ball, studies their growth, and analyzes the boundedness and compactness of associated weighted composition operators.

Contribution

It provides necessary and sufficient conditions for Hadamard gap series to be in weighted-type spaces and characterizes the boundedness and compactness of related operators.

Findings

Characterization of functions with Hadamard gaps in weighted-type spaces.

Growth rate analysis of functions in these spaces.

Criteria for boundedness and compactness of weighted composition operators.

Abstract

We give a sufficient and necessary condition for an analytic function $f(z)$ on the unit ball $\BB$ in $\CC^n$ with Hadamard gaps, that is, for $f(z)=\sum_{k=1}^\infty P_{n_k}(z)$ where $P_{n_k}(z)$ is a homogeneous polynomial of degree $n_k$ and $n_{k+1}/n_k \ge c>1$ for all $k \in \NN$, to belong to the weighted-type space $H^\infty_\mu$ and the corresponding little weighted-type space $H^\infty_{\mu, 0}$, under some condition posed on the weighted funtion $\mu$. We also study the growth rate of those functions in $H^\infty_\mu$. Finally, we characterize the boundedness and compactness of weighted composition operator from weighted-type space $H^\infty_\mu$ to mixed norm spaces.